Motivated by the recent angle-resolved photoemission spectroscopy (ARPES) measurements on one-dimensional Mott insulators, SrCuO2 and Na 0.96V2O5, we examine the single-particle spectral weight of the one-dimensional (1D) Hubbard model at half-filling and in the doped case. We are particularly interested in the temperature dependence of the spinon and holon excitations. For this reason, we have performed dynamical density matrix renormalization group and determinantal quantum Monte Carlo (QMC) calculations for the single-particle spectral weight of the 1D Hubbard model. In the QMC data, the spinon and holon branches become observable at temperatures where the short-range antiferromagnetic correlations develop. At these temperatures, the spinon branch grows rapidly. In the light of the numerical results, we discuss the spinon and holon branches observed by the ARPES experiments on SrCuO2. These numerical results are also in agreement with the temperature dependence of the ARPES results on Na0.96V 2O5. In addition, we briefly discuss the spectral weight in the doped case.